Abstract
We apply the homotopy perturbation Sumudu transform method (HPSTM) to the time-space fractional coupled systems in the sense of Riemann-Liouville fractional integral and Caputo derivative. The HPSTM is a combination of Sumudu transform and homotopy perturbation method, which can be easily handled with nonlinear coupled system. We apply the method to the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The simplicity and validity of the method can be shown by the applications and the numerical results.
1. Introduction
Fractional calculus, compared to integer calculus, was mentioned in a letter from L’Hospital to Leibniz in 1695. In the letter, L’Hospital raised a question, “what is the result of if ?” The answer of Leibniz was “ will be equal to . This is an apparent paradox, from which, one day useful consequences will be drawn” [1]. Furthermore, the generalization of this framework indicates that it is more appropriate to talk about integration and differentiation of arbitrary order, such as fractional order, real number order, and even complex number order just as the development of number system. Thus, there is a basic question: “what are the definitions of fractional integral and derivative?” Or “how to define the fractional integral and derivative?” More and more mathematicians focused on this problem, like Lagrange, Laplace, Fourier, and so on. Some different fractional integrals and derivatives have been given according to different needs, like Riemann-Liouville fractional integral, Caputo fractional derivative, Weyl fractional derivative, and so on [2]. But there are no uniform definitions of fractional integral and derivative, and the frequently used definitions are Riemann-Liouville integral and Caputo derivative.
Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so on, which involve derivatives of fractional order. Fractional differential equations also serve as an excellent tool for the description of hereditary properties of various materials and processes. In consequence, the subject of fractional differential equations is gaining much more attention. For example, in electromagnetism, Sebaa et al. [3] studied ultrasonic wave propagation in human cancellous bone by using fractional calculus to describe the viscous interactions between fluid and solid structure. In signal processing, Assaleh and Ahmad [4] proposed a new approach for speech signal modeling through using fractional calculus. Magin and Ovadia [5] molded the cardiac tissue electrode interface using fractional calculus. In control theory, Suarez et al. [6] applied fractional controllers to the path-tracking problem in an autonomous electric vehicle. In fluid mechanics, Kulish and Lage [7] applied fractional calculus to the solution of time-dependent, viscous-diffusion fluid mechanics problems.
In this paper, we intend to construct the approximate solutions to the nonlinear time-space fractional coupled systems. There are many effective methods to solve this problem, like Adomian decomposition method [8–10], variation iteration method [11], differential transform method [12], residual power series method [13, 14], iteration method [15], homotopy perturbation method [16], homotopy analysis method [17], and so on. Furthermore, for the nonlinear problem, the multiple exp-function method [18, 19], the transformed rational function method [20–22], and invariant subspace method [23, 24] are three systematical approaches to handle the nonlinear terms. The first one is to propose the exact solution of nonlinear partial differential equations by using rational function transformations. Its key point is to search for rational solutions to variable-coefficient ordinary differential equations transformed from given partial differential equations. The second one is to consider the form of solution as rational exponential functions with unknown coefficients whose advantage is direct applicability to underlying equation. The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. Motivated by these fruitful results, Singh et al. [25] proposed the homotopy perturbation Sumudu transform method based on the homotopy perturbation method and Sumudu transform method and applied it to nonlinear partial differential equations. The HPSTM was extended to the time-fractional PDEs in [26, 27]. It is worth mentioning that the HPSTM is applied without any using of Adomian polynomials, over restrictive assumption or linearization, and is capable of reducing the volume of computational work as compared to the classical numerical methods while still maintaining the high accuracy of the result. Meanwhile, it is appropriate not only for strongly nonlinear system but also for weakly nonlinear system.
The rest of the paper is organized as follows. In Section 2, we introduce some concepts on fractional calculus and the Sumudu transform. In Section 3, we illustrate the basic idea of HPSTM which is applied to the time-space fractional coupled systems. In Section 4, we apply HPSTM to obtain fractional power series solutions of nonlinear time-space fractional coupled systems with initial values, and some numerical results are presented as well.
2. Preliminaries
Definition 1 (see [2]). A real function , is said to be in the space , , if there exists a real number , such that , where , and it is said that , if , .
Definition 2 (see [2]). The fractional integral of in the Riemann-Liouville (left-sided) sense is defined aswhere , , , and is the Gamma function.
Definition 3 (see [2]). The fractional integral of in the Riemann-Liouville sense is defined aswhere , , , and is the Gamma function.
Definition 4 (see [2]). The Caputo (left-sided) fractional derivative operator of order , of a function , is defined as
Definition 5 (see [2]). The Caputo fractional derivative operator of order , of a function , is defined as
Lemma 6 (see [28]). If , , and , , one has
In 1998, a new integral transform, named Sumudu transform, was introduced by Watugala [29] to study solutions of ordinary differential equations in control engineering problems. The Sumudu transform is defined over the set of functions by the following formula:
Property 7 (see [30]). (i) The Sumudu transform satisfies linear property; that is,where , are constants.
(ii)
Lemma 8 (see [31]). The Sumudu transform of the Caputo fractional derivative is
3. Homotopy Perturbation Sumudu Transform Method
In this section, to illustrate the basic idea of this method, we consider a general nonhomogeneous fractional partial differential coupled systemwith the initial conditionswhere , , , , denote linear differential operators and nonlinear differential operators, respectively, and are the source terms. Applying the Sumudu transform on both sides of (10) yields It follows from the property of the Sumudu transform in (9) thatFurthermore, applying the inverse Sumudu transform on both sides of (13) yields where , , represent the terms arising from the source terms and prescribed initial conditions; that is,
Let us construct the homotopy perturbation equationswhere homotopy parameter Suppose that and the nonlinear terms can be written aswhere the coefficient polynomials can be determined below and are given by the following formulae:Substituting (17) into (16) gives Comparing the coefficients of , we obtain the following recurrence equations:where .
According to the homotopy perturbation method, we assume that the solution of (10)-(11) can be written asSetting , the approximate solution to (10)-(11) isThe convergence of series (21) depends on the nonlinear differential operator . Generally, the derivative with respect to of the nonlinear part in the splitting must be sufficiently small, since the parameter may be relatively large; in fact we take . The series is convergent for most cases [32].
Remark 9. HPSTM is applied to construct homotophy series solutions for fractional coupled systems, which has not too many overstrict assumptions compared to some classical methods.
4. Application of HPSTM to Time-Space Fractional Coupled Systems
In this section, we apply HPSTM to nonlinear time-space fractional coupled systems with initial conditions.
4.1. The Time-Space Fractional Coupled Burgers System
The Burgers equation is one of the most important partial differential equations from fluid mechanics, which not only describes many phenomena, for example, modeling the motion of turbulence [33], but also has many applications in science and engineering [34]. Here we apply HPSTM to solve the following nonlinear time-space fractional coupled Burgers system:with the initial conditions where , .
Applying the Sumudu transform on both sides of (23) with the initial conditions, we can obtainThe inverse Sumudu transform of (25) implies thatNow applying the homotopy perturbation method gives where and are polynomials which denote the homotopy coefficients of the nonlinear term and are given by Set and then Comparing the coefficients of , this gives Generally, we havewhere Hence, the series solution of (23) is Particularly, when , the exact solution of (23) is Using HPSTM, when , the third approximate solution of (23) is In general, the limit of the approximate solution is which is as same as the exact solution. However, if the initial values are too complex to find the limit of the approximated solution, then we replace the exact solution by the approximated solution within a certain scale, which is useful in the application of engineering.
Thus we plot the images of the approximate solution (see Figure 1(a)), the exact solution (see Figure 1(b)), and the error function (see Figures 1(c) and 1(d)). It is clear that the error function depends on time . When time is small (e.g., ), the error function is in the scale of (see Figure 1(c)), which indicates that this is a good approximation in the neighbour of time for system (23) with some explicit parameters. However, when time becomes large (e.g., ), the error function tends to be large as well (see Figure 1(d)); that is to say, this method is only suitable for constructing the approximated solution around the initial data.
(a) The third-order approximate solution of
(b) The exact solution of
(c) The error of at
(d) The error of at
4.2. The Time-Space Fractional Coupled KdV System of Generalized Hirota-Satsuma Type
In this subsection, consider the time-space fractional generalization of the Hirota-Satsuma coupled KdV systemwith respect to the initial conditions where , , , , . The Hirota-Satsuma coupled KdV equation describes the unidirectional propagation of shallow water waves, which was initiated by Wu et al. [35]. Further (38) becomes a generalized fractional KdV equation for and a fractional MKdV equation for
Applying the Sumudu transform on both sides of (38) with the initial conditions, we obtainThe inverse Sumudu transform of (40) implies thatVia the homotopy perturbation method, it giveswhere , , and are polynomials which denote the nonlinear term, and they are given by Set and then Comparing the coefficients of showsGenerally, one haswhereTherefore, the approximate series solution of (38) is Particularly, when , , and the special initial value of (38) is then the exact solutions are Under these special conditions, via HPSTM, the first approximate solution of (38) is Similarly, we obtain the following numerical results: see Figures 2(a), 2(b), 2(c), 2(d), 3(a), 3(b), 3(c), 3(d), 4(a), 4(b), 4(c), and 4(d).
(a) The first-order approximate solution of
(b) The exact solution of
(c) The error of at
(d) The error of at
(a) The first-order approximate solution of
(b) The exact solution of
(c) The error of at
(d) The error of at
(a) The first-order approximate solution of
(b) The exact solution of
(c) The error of at
(d) The error of at
4.3. The Time-Space Fractional Coupled Shallow Water System
Shallow water systems are widely used in many areas of fluid dynamics, such as multiphase flows [36], turbulence [37], and viscoelasticity [38]. It is well known that the shallow water systems can accurately predict both the hydraulic parameters under conditions of slow erosion and low sediment concentration. Let us consider the time-space fractional coupled shallow water systemwith initial valueswhere , , , , ,
Applying the Sumudu transform on both sides of (53) with the initial conditions, we obtainThe inverse Sumudu transform of (55) implies thatAccording to homotopy perturbation method, we havewhere and are polynomials of the nonlinear term and are given by Setting then we arrive at Comparing the coefficients of yieldsGenerally, we have whereHence, the series solution is
4.4. The Time-Space Fractional Coupled KdV System
KdV equation plays an important role in nonlinear equations for wide applications in physics and engineering. Hirota and Satsuma [39] firstly found coupled KdV system to describe the iterations of water waves; meanwhile, they claimed that the system exists with a soliton solution. In [40], Fan and Zhang settled several kinds of solutions by an improved homogeneous method. The time-space fractional coupled KdV equation is a generalization of the classical coupled KdV equation. In this subsection, we consider the following time-space fractional coupled KdV system:with respect to initial valueswhere , , , , and the coefficients , are constants.
Applying the Sumudu transform on both sides of (65) with the initial conditions, we obtainThe inverse Sumudu transform of (67) implies thatAnalogously, using homotopy perturbation method gives where and are homotopy polynomials coefficients of the nonlinear term, which are given by Setting then Comparing the coefficients of , Generally, we get whereThus, the series solution of (65) is
4.5. The Time-Space Fractional Coupled Whitham-Broer-Kaup (WBK) System
Under the Boussinesq approximation, Whitham [41], Broer [42], and Kaup [43] obtained the following nonlinear WBK system. In this subsection, we construct the approximate solution by the HPST method to the time-space fractional coupled WBK system.
Consider the time-space fractional coupled WBK systemwith respect to the initial conditions where , , , , , denote different dispersive power, is the field of horizontal velocity, and is the height deviating equilibrium position of liquid.
Applying the Sumudu transform on both sides of (77) with the initial conditions, we obtainThe inverse Sumudu transform of (79) implies thatUsing homotopy perturbation method, it leads towhere and are polynomials of the nonlinear term and are given by Setting then Comparing the coefficients of , this givesGenerally, we have where Hence, the series solution of (77) is
Remark 10. When , , and , (77) reduces to the classical long-wave system that describes the shallow water wave with diffusion.
Remark 11. When , , and , (77) reduces to the variant Boussinesq system.
5. Concluding Remarks
In this paper, we apply the HPSTM to the nonlinear time-space fractional coupled equations. Applying the HPSTM, we can obtain analytic and approximate solutions to different coupled systems, for example, the coupled Burgers system, the coupled KdV system, the generalized Hirota-Satsuma coupled KdV system, the coupled WBK system, and the coupled shallow water system. The advantage of the HPSTM is its capability of combining two powerful methods for obtaining exact and approximate analytical solutions for nonlinear system. It provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, perturbation, or restrictive assumptions. The numerical results indicate that this method is effective and simple in constructing analytic or approximate solutions to fractional coupled systems.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors thank Dr. Yixian Gao for many useful suggestions and help. This research was supported in part by NSFC Grants 201114110, 11571065, 11671071, and 11401089, NSFJL Grants 20160520094JH and 20170101044JC, EDJLP Grant TTKH20170904KJ, and the Fundamental Research Funds for the Central Universities, 2412017FZ005.